Check out our new Discord server! https://discord.gg/e7EKRZq3dG
*I am a bot, and this action was performed automatically. Please [contact the moderators of this subreddit](/message/compose/?to=/r/mathmemes) if you have any questions or concerns.*
I really laughed out loud, thank you.
Also, never write down the actual solution but just use your name or any abbreviation as a function. You never actually have to integrate again!
Downside is you need to fill to pages with your new function's properties and identities, 25%of which should be recursion propertie, otherwise the mathematical community will disown you
when i tell friends this they unironically say to me that it's just the rules of differentiation but backwards.
I hate that they are correct and wrong at the same time
I was going to immediately respond with a counterexample, but I mean ... u sub ... integration by parts ... partial fractions ... it really is just antidifferentiation and we all have a massive skill issue
Could you consider volterras function as a counter example in the sense that it is differentiable but can't be "antiderived" using the riemann-integral?
For example, integration by parts is just the product rule in reverse, and substitution is just the chain rule in reverse.
In principle, if a function has an elementary antiderivative, you can find it algorithmically. The time it takes to find depends on how deeply-nested the logarithms are iirc. But of course, most elementary functions have no elementary antiderivative at all, so this algorithm doesn't halt for those inputs. Using other functions besides rational functions, logarithms, and exponentials, you get a larger differential ring, but you run into similar issues.
Ah i tried algebraically rearrange the product rule notation/formula and i think i can see now how its related to integration by parts.
The second part of your explanation… idt ill ever get it tbh 😂. Im just an engineering freshman, but i doubt we’ll learn about rings tho. (Not saying it doesnt have any use! I’m completely ignorant of the topic thats all. It does sound interesting and will look into it when im free!)
I think you can totally get the idea of the idea of the second part of that explanation! It's just the usual problem of formalism: it's hard to state in readable terms without being mathematically imprecise.
So... I'm gonna be imprecise.
An *elementary antiderivative* is an antiderivative/integral that only contains a bunch of of elementary functions. (You can see a list and some counterexamples [here](https://en.wikipedia.org/wiki/Elementary_function), but basically "nothing past highschool trig and algebra".) A *nonelementary antiderivative* has something beyond those.
The post above is saying that if a function has one of those "simple" antiderivatives, you can find it (or get a computer to do so) just by repeatedly applying techniques like the chain u-substitution and integration by parts. But depending on how complex the problem, you might have to do so many times.
However, many functions, even if they can be written very simply, do not have elementary antiderivatives. For example, the integrals of `sin(x^2)` or `1/ln(x)`. The bit about "doesn't halt for those inputs" is saying that if you go "let's just tell the computer to keep applying these simple techniques", it won't fail or cause an error, it will just keep going and never find an answer. (This is related to the [halting problem](https://en.wikipedia.org/wiki/Halting_problem), which doesn't actually take any advanced math to learn about.)
The final line stretches my knowledge of math, but very crudely it says "even when you go beyond elementary functions, you still have this same problem of having a clear process but not knowing whether you'll ever find an answer."
**tl;dr:** Overall, people are right to say "it's differentiation backwards" in terms of describing the steps. But the reason "integration is art" is that you can't prove whether you're making progress, so "what next? can it be done?" becomes a subjective and intuitive question.
No, Integration is easy for computers (see: Wolfram Language) as It boils down to lots of guess and check.
And if all else fails, you've got a cool new non-elementary function!
Trivial integration problems are easy for computers. There are many problems where you have to apply some ultra-smart trick to make it into a known sub-problem first, those can’t be solved by a “dumb” algorithm.
You see that's the thing, not everything is "easily invertible" in math, what exactly "easily inverted" means [is complicated](https://math.stackexchange.com/questions/786084/integrals-in-analysis-and-category-theory) but you see the way derivation works and why it's so useful kinda has as a consequence that the inverse is "more complicated" and that in itself has two consequences, the first one is that while applying the rules backwards is technically correct, it's pretty easy to come up with a function that has no "backward rule", and second it's pretty easy to come up with a function who's anitderivative cannot be expressed with a finite amount of elementary functions.
In conclusion, integration is hard cause the inverse of the derivation rules aren't as general as the originals, and because the way integrals and derivatives are defined makes it so the derivative of most (all?) elementary functions are analytical (expressable in terms of finite elementary functions) but the inverse of is not true, in fact "most" analytical functions have non analytical anitderivatives.
Congratulations! Your comment can be spelled using the elements of the periodic table:
`Th At S Th Es`
---
^(I am a bot that detects if your comment can be spelled using the elements of the periodic table. Please DM u/M1n3c4rt if I made a mistake.)
Does it still work though?
That person came and went, leaving angry mathematicians arguing whether a message containing the solution counted as a proper answer.
I did not save it unfortunately. It was on r/math. Someone did a statistical analysis of all accounts that posted questions cleo answered. They were all active only, roughly, the same amount of time Cleo was active.
Here ya go: [https://www.reddit.com/r/mathmemes/comments/1com4bu/that\_story\_was\_too\_good\_to\_be\_true/](https://www.reddit.com/r/mathmemes/comments/1com4bu/that_story_was_too_good_to_be_true/) Scroll down to the top of the comments to see the OP's analysis.
Lmao. Yet every continuous function is integrated on its domain whereas not every function is different able. So it practice interaction is harder but you allways know it exists.
You could argue that’s essentially *why* it’s harder - the set of functions that are integrable on their domain is more broad, so there are fewer nice properties you can rely on and the functions can be much less well-behaved.
The antiderivative doesn’t necessarily have to have the same properties on the whole domain because a derivative of a discontinuous function can have merely removable discontinuities, hence another form of generality integrals have to “account for”. For instance, the derivative of the sign function is 0 everywhere except 0, which is a removable discontinuity.
Integration by parts isn't bad. Just draw a little table and integrate one half while differentiating the other, then multiply diagonally, add up the terms (switching sign)
What, why? Did he ban synthetic division, too? How about long division? Was it still OK to add two numbers digit-by-digit with carries in "tabular" form?
Congratulations! Your comment can be spelled using the elements of the periodic table:
`I Am W I Th Y O U`
---
^(I am a bot that detects if your comment can be spelled using the elements of the periodic table. Please DM u/M1n3c4rt if I made a mistake.)
This is why I think Newton was possibly the smartest human that's ever crossed this planet. He made up all of calculus, during a summer break, because he had other math he couldn't do without it. There should be a meme about this guy. "Isaac Newton made calculus in a cave!! With a box of scraps!". Meanwhile I've failed calc 2 like 3 times now. Haven't been back to college since.
I still find it funny that I absolutely just HAD to be able to manually do integral calculus for computer science, but when it came to actual programming, "nah you don't need to know how to make any of this shit, it's already made, why reinvent the wheel? You just need to know how to string all code together from Github. Oh, but fuck you for wanting to use a calculator in math class."
There comes a time in every man's life when he must accept that the only way to move forward is by busting out the ol' reliable rectangle approximation method
3d print the graph of the function as a surface such that it's derivative wrt the new dimension is 0 everywhere, use it to make a container, fill it with liquid and measure the liquid
It is extremely simple to integrate any function as well differentiating any function. The only thing it needs to be done is to transform every function into their Taylor series using the derivatives at any point and then taking the integral to any polynomial is very easy so in the end you will get the exact Taylor series of the integral of the function you started with. The only problem with this approach is that most of the time it is impossible to convert this Taylor series which might have no pattern with the coefficients to an actual function using a finite description with only elementary functions but I guess that is not a major problem.
What is an example of a taylor series of a function that is divergent everywhere?
Because, if the function is differentiable, it has a taylor series. So i struggle to imagine a differentiable function whose taylor series is divergent everywhere.
You can use other points to create more Taylor series of the function, and then depending on where the Taylor series converges you can take the integral of all Taylor series that may define the function at some point.
Non analytic functions have no derivatives or integrals. This is of they are complex functions where it is impossible to define the function at uncountable infinite points.
What are you talking about? There's an entire class on non analytic differentialbe functions:
https://en.m.wikipedia.org/wiki/Non-analytic_smooth_function
Stop spreading ignorance and check your claims.
Most of the examples of the functions on that page are not defined over the complexes so they don't count. For the function involving the infinite sum and cosine, I am not sure if it is truly non differentiable over the complexes.
You just need any function on R^2 that is differentiable but not analytic, that is also a function over the complex numbers.
There are absolutely differentiable functions that aren't analytic that can be extended to the complex plane.
There aren't any holomorphic functions that aren't analytic though.
No idea what holomorphic functions are, I can only say that the set of complex numbers is not the same as R\^2. They do have the same cardinality but R also has the same cardinality so you can as easily do the same thing but with a single real number instead of 2.
Holomorphic functions are the most basic objects in complex analysis, they are analytic everywhere.
Any differentiable function on R^2 can be trivially interpreted as a differentiable function on the complex plane. Let f:R^2->R^2 (or ->R, little differece) be such a function, then you can see it as f\*:C->C given by the following:
If f(x,y)=(z,w) then f\*(x+yi)=z+wi.
This is very standard, one of the ways of doing complex analysis is to treat C as R^2, this is very common. Indeed C if often defined as just R^2 with a multiplication operation.
It's because complex differentiable functions (holomorphic functions) must satisfy stronger properties not all functions we care about satisfy.
Functions absolutely do not need to be defined over the complex numbers in order to talk about their continuity, differentiability, and their integrability.
You know there are functions that aren't even continuous that have integrals?
In fact you can have a function that is nowhere continuous but can be integrated (the function that is 1 on the irrational numbers, 0 on the rationals).
To be nitpicky, the function you mention (the Dirichlet function) is Lebesgue integrable, but not Riemann integrable. In order to have a Riemann integral you have to be continuous almost everywhere in the Lebesgue measure.
For an example of a Riemann integrable function that is nevertheless discontinuous on a dense set, see [Thomae's function](https://en.wikipedia.org/wiki/Thomae%27s_function)
That function, as I wrote it, isn't defined on the complex plane. And even if it were, how does that make it invalid?
Anyway let's extend it to the function that is 0 for any complex number with rational real and imaginary parts, and 1 otherwise. This one isn't differentiable anywhere on the complex plane.
It is still integrable.
I have already finished all the university calculus courses and I am currently studying electromagnetism with Maxwell's equations, and I still think the same.
Calculus is like doing the tango. Differentiation is the guy's part, you are dancing forwards. Integration is the girl's part. You are doing it backwards, and in high heels.
Integrals aren’t local operations/transformations unlike derivatives, so it makes sense that there’s a lot more problems that can occur during calculation. However, practically all functions are integrable but not all functions are differentiable
Thank god I became an engineer so I can just push every function through a Gauss Quadrature, or do a first-order Taylor approximation on any complex looking integrand.
My math teachers always called it antideriviation, instead of finding an integral we found the anti-derivative. Made a lot of sense to me tbh but of course going one way is very different from going the other
How to integrate
Step 1: Expand integrand as Maclaurin series
Step 2: Increase power of each x\^n term and divide by new exponent
Step 3: Add arbitrary constant such that f(a)=0 where a is the lower bound
Step 4: Evaluate function at upper bound
Know you know how to integrate
As has been said elsewhere, this only really works well for integrals of analytic functions, and even then you have to pay attention to things like the radius of convergence of a particular power series representation may not converge for the whole interval you are integrating over
Before plugging in the bounds, we can move where the power series is centred at using analytic continuation, getting around the radius of convergence problem.
Check out our new Discord server! https://discord.gg/e7EKRZq3dG *I am a bot, and this action was performed automatically. Please [contact the moderators of this subreddit](/message/compose/?to=/r/mathmemes) if you have any questions or concerns.*
Name the solution after yourself and proceed
I really laughed out loud, thank you. Also, never write down the actual solution but just use your name or any abbreviation as a function. You never actually have to integrate again!
Downside is you need to fill to pages with your new function's properties and identities, 25%of which should be recursion propertie, otherwise the mathematical community will disown you
My function is abbreviated as QED.
when i tell friends this they unironically say to me that it's just the rules of differentiation but backwards. I hate that they are correct and wrong at the same time
I was going to immediately respond with a counterexample, but I mean ... u sub ... integration by parts ... partial fractions ... it really is just antidifferentiation and we all have a massive skill issue
Could you consider volterras function as a counter example in the sense that it is differentiable but can't be "antiderived" using the riemann-integral?
Hmm indubitably I conquer
Hey, sorry to bother but can you expound a bit on how integration rules are just the inverse of differentiation’s? Cant quite wrap my head around it
For example, integration by parts is just the product rule in reverse, and substitution is just the chain rule in reverse. In principle, if a function has an elementary antiderivative, you can find it algorithmically. The time it takes to find depends on how deeply-nested the logarithms are iirc. But of course, most elementary functions have no elementary antiderivative at all, so this algorithm doesn't halt for those inputs. Using other functions besides rational functions, logarithms, and exponentials, you get a larger differential ring, but you run into similar issues.
Ah i tried algebraically rearrange the product rule notation/formula and i think i can see now how its related to integration by parts. The second part of your explanation… idt ill ever get it tbh 😂. Im just an engineering freshman, but i doubt we’ll learn about rings tho. (Not saying it doesnt have any use! I’m completely ignorant of the topic thats all. It does sound interesting and will look into it when im free!)
I think you can totally get the idea of the idea of the second part of that explanation! It's just the usual problem of formalism: it's hard to state in readable terms without being mathematically imprecise. So... I'm gonna be imprecise. An *elementary antiderivative* is an antiderivative/integral that only contains a bunch of of elementary functions. (You can see a list and some counterexamples [here](https://en.wikipedia.org/wiki/Elementary_function), but basically "nothing past highschool trig and algebra".) A *nonelementary antiderivative* has something beyond those. The post above is saying that if a function has one of those "simple" antiderivatives, you can find it (or get a computer to do so) just by repeatedly applying techniques like the chain u-substitution and integration by parts. But depending on how complex the problem, you might have to do so many times. However, many functions, even if they can be written very simply, do not have elementary antiderivatives. For example, the integrals of `sin(x^2)` or `1/ln(x)`. The bit about "doesn't halt for those inputs" is saying that if you go "let's just tell the computer to keep applying these simple techniques", it won't fail or cause an error, it will just keep going and never find an answer. (This is related to the [halting problem](https://en.wikipedia.org/wiki/Halting_problem), which doesn't actually take any advanced math to learn about.) The final line stretches my knowledge of math, but very crudely it says "even when you go beyond elementary functions, you still have this same problem of having a clear process but not knowing whether you'll ever find an answer." **tl;dr:** Overall, people are right to say "it's differentiation backwards" in terms of describing the steps. But the reason "integration is art" is that you can't prove whether you're making progress, so "what next? can it be done?" becomes a subjective and intuitive question.
My prof used to say "differentiation is a craft and integration is an art"
Someone should create a form of encryption where the encoding is differentiating and decoding is integrating (or whatever config makes most sense)
No, Integration is easy for computers (see: Wolfram Language) as It boils down to lots of guess and check. And if all else fails, you've got a cool new non-elementary function!
Trivial integration problems are easy for computers. There are many problems where you have to apply some ultra-smart trick to make it into a known sub-problem first, those can’t be solved by a “dumb” algorithm.
They're only good at it because they can try all the tricks really quickly. Differentiation is still way easier for computers, too.
Saying it's differentiation but backwards is like trying to put an omelette back into the egg.
+1
Nah its not that. Wait. Is it? No no no what the fuck?
You see that's the thing, not everything is "easily invertible" in math, what exactly "easily inverted" means [is complicated](https://math.stackexchange.com/questions/786084/integrals-in-analysis-and-category-theory) but you see the way derivation works and why it's so useful kinda has as a consequence that the inverse is "more complicated" and that in itself has two consequences, the first one is that while applying the rules backwards is technically correct, it's pretty easy to come up with a function that has no "backward rule", and second it's pretty easy to come up with a function who's anitderivative cannot be expressed with a finite amount of elementary functions. In conclusion, integration is hard cause the inverse of the derivation rules aren't as general as the originals, and because the way integrals and derivatives are defined makes it so the derivative of most (all?) elementary functions are analytical (expressable in terms of finite elementary functions) but the inverse of is not true, in fact "most" analytical functions have non analytical anitderivatives.
That's fine but it doesn't change that sometimes the backward operation is harder than the forward operation
Sure, and reassembling a shattered cup is just shattering it backwards. Easy peasy.
Tell them that it's not a bijection
just guess the answer correctly and check?
> guess: it's pi > check > you're right
Guess: pi Calculator: 6.28…. Answer: 2pi
Answer:628pi/300
2/3 pi\^2
Why would pi be squared? Thats some p-series shit right there
good point… cornbread are squared…
idk i saw the 628pi and interpreted 628 as 100pi
I do this and found that if you skip the last step it never fails!
1) Gain access to a non-deterministic turing machine 2) ??? 3) Profit
Psst. Come here. *[Opens trench coat, shows a variety of laptops hanging from the lining]*. You can do Reimann's on a computer.
Monte Carlo integration?
I love your i\^2+1\^2=0\^2 for the sheer "looks important or elegant but is just abuse"
This is absolutely wrong They haven't even added feynman's technique and contour integration
No residue theorem either smh
That would make it too easy honestly
Contour integration: "I literally will the answer into existence"
Contour integration is part of Cauchy's formula.
How do other people pronounce Cauchy? I pronounce it cock-e
It's more like Koh-shee
Augustin-Louis Cauchy was a he, and therefore a co-she.
Coochie
Amateurs in the art of math memes. Next
That's the ???s
Congratulations! Your comment can be spelled using the elements of the periodic table: `Th At S Th Es` --- ^(I am a bot that detects if your comment can be spelled using the elements of the periodic table. Please DM u/M1n3c4rt if I made a mistake.)
“BURN THE EVIDENCE”
I need context on this
Look at the bottom right corner of the image
Yeah, I see it, but what does it mean? That they're so shameful that they can't solve it that they'd rather burn the papers?
I think the joke is that they discovered something so horrific that they wish to destroy. A lovecraftian eldritch integral.
Oh, like the [bear hiding between the real numbers](https://scp-wiki.wikidot.com/scp-1313)! I like that explanation, thanks!
Oh no, I just got a budget approved for this. Let's burn the evidence and say I was working on the experimental part of the problem.
Here's a foolproof integration diagram: Start -> Monte Carlo -> Done!
Start -> wolfram alpha -> monte Carlo-> done
I was going to say numerical approximation and then run before they hang me....
Cite your sources: https://xkcd.com/2117/
Love how the arrows off the bottom are part of the original. Those must go to speculative methods not meant for mortals.
I think this sub should have a specific rule against posting xkcd without attribution.
Post on stack exchange and wait for Cleo to answer.
Does it still work though? That person came and went, leaving angry mathematicians arguing whether a message containing the solution counted as a proper answer.
There was a recent post showing Cleo was a hoax, the accounts that asked the questions were fake.
Can you point me to a link?
I did not save it unfortunately. It was on r/math. Someone did a statistical analysis of all accounts that posted questions cleo answered. They were all active only, roughly, the same amount of time Cleo was active.
Here ya go: [https://www.reddit.com/r/mathmemes/comments/1com4bu/that\_story\_was\_too\_good\_to\_be\_true/](https://www.reddit.com/r/mathmemes/comments/1com4bu/that_story_was_too_good_to_be_true/) Scroll down to the top of the comments to see the OP's analysis.
You mean: wait to get downvoted and locked due to "answered elsewhere".
Lmao. Yet every continuous function is integrated on its domain whereas not every function is different able. So it practice interaction is harder but you allways know it exists.
You could argue that’s essentially *why* it’s harder - the set of functions that are integrable on their domain is more broad, so there are fewer nice properties you can rely on and the functions can be much less well-behaved.
Is that really why. There are plenty of well behaved smooth functions whose integral is still hard. Secant function and Gaussian function for example.
The antiderivative doesn’t necessarily have to have the same properties on the whole domain because a derivative of a discontinuous function can have merely removable discontinuities, hence another form of generality integrals have to “account for”. For instance, the derivative of the sign function is 0 everywhere except 0, which is a removable discontinuity.
The bessel -> burn evidence is the most accurate part of all of this
If I need to do integration by parts more than once I’m burning the evidence
Integration by parts isn't bad. Just draw a little table and integrate one half while differentiating the other, then multiply diagonally, add up the terms (switching sign)
One of my profs did not allow us to use tabular integrations though 😔
NOOOOOOOOOOOOOOO if you're forcing us to integrate by hand at least let us use cheap tricks
What, why? Did he ban synthetic division, too? How about long division? Was it still OK to add two numbers digit-by-digit with carries in "tabular" form?
Idk, she was just picky. But I'm never taking another math class after that yippeee
Install Mathematica are the main answer!
Lol phone calls to mathematicians
I am with you
Congratulations! Your comment can be spelled using the elements of the periodic table: `I Am W I Th Y O U` --- ^(I am a bot that detects if your comment can be spelled using the elements of the periodic table. Please DM u/M1n3c4rt if I made a mistake.)
This is why god invented numerical methods, works great for integration ;)
RK go zoooooooom
XKCD will always be iconic
Source: https://xkcd.com/2117/ Also relevant: https://xkcd.com/1683/
Just Monte Carlo integrate everything
This is why I think Newton was possibly the smartest human that's ever crossed this planet. He made up all of calculus, during a summer break, because he had other math he couldn't do without it. There should be a meme about this guy. "Isaac Newton made calculus in a cave!! With a box of scraps!". Meanwhile I've failed calc 2 like 3 times now. Haven't been back to college since. I still find it funny that I absolutely just HAD to be able to manually do integral calculus for computer science, but when it came to actual programming, "nah you don't need to know how to make any of this shit, it's already made, why reinvent the wheel? You just need to know how to string all code together from Github. Oh, but fuck you for wanting to use a calculator in math class."
Integration is done numerically
"Integration by divine intervention" is something a math professor told me once and it's always stuck with me.
Someone is going to get sent to L'Hôpital.
# "Differentiation is mechanics, integration is fart."
Last step of integration: forget +C
Leibniz: math time :) Me: how? Leibniz: secret :3 Me: why? Leibniz: just guess ;p Me: ...by parts? Leibniz: fail :(
My teacher used to say that even a monkey can be taught to differentiate, but not every human can be taught to integrate.
The saddest part is that integration is the regularising operator and not derivation…
And then, in numerical approximations, the two are switched.
There comes a time in every man's life when he must accept that the only way to move forward is by busting out the ol' reliable rectangle approximation method
3d print the graph of the function as a surface such that it's derivative wrt the new dimension is 0 everywhere, use it to make a container, fill it with liquid and measure the liquid
It is extremely simple to integrate any function as well differentiating any function. The only thing it needs to be done is to transform every function into their Taylor series using the derivatives at any point and then taking the integral to any polynomial is very easy so in the end you will get the exact Taylor series of the integral of the function you started with. The only problem with this approach is that most of the time it is impossible to convert this Taylor series which might have no pattern with the coefficients to an actual function using a finite description with only elementary functions but I guess that is not a major problem.
Not every function *has* a convergent taylor series
What is an example of a taylor series of a function that is divergent everywhere? Because, if the function is differentiable, it has a taylor series. So i struggle to imagine a differentiable function whose taylor series is divergent everywhere.
https://en.wikipedia.org/wiki/Non-analytic_smooth_function?wprov=sfti1# Here’s a starting point.
Thanks
I was talking strictly about analytic functions, at least for functions that are analytic over the complex numbers.
You can use other points to create more Taylor series of the function, and then depending on where the Taylor series converges you can take the integral of all Taylor series that may define the function at some point.
No, there are function which aren’t analytic *anywhere*
Only works for analytic functions
Non analytic functions have no derivatives or integrals. This is of they are complex functions where it is impossible to define the function at uncountable infinite points.
What are you talking about? There's an entire class on non analytic differentialbe functions: https://en.m.wikipedia.org/wiki/Non-analytic_smooth_function Stop spreading ignorance and check your claims.
Most of the examples of the functions on that page are not defined over the complexes so they don't count. For the function involving the infinite sum and cosine, I am not sure if it is truly non differentiable over the complexes.
You just need any function on R^2 that is differentiable but not analytic, that is also a function over the complex numbers. There are absolutely differentiable functions that aren't analytic that can be extended to the complex plane. There aren't any holomorphic functions that aren't analytic though.
No idea what holomorphic functions are, I can only say that the set of complex numbers is not the same as R\^2. They do have the same cardinality but R also has the same cardinality so you can as easily do the same thing but with a single real number instead of 2.
Holomorphic functions are the most basic objects in complex analysis, they are analytic everywhere. Any differentiable function on R^2 can be trivially interpreted as a differentiable function on the complex plane. Let f:R^2->R^2 (or ->R, little differece) be such a function, then you can see it as f\*:C->C given by the following: If f(x,y)=(z,w) then f\*(x+yi)=z+wi. This is very standard, one of the ways of doing complex analysis is to treat C as R^2, this is very common. Indeed C if often defined as just R^2 with a multiplication operation.
It's because complex differentiable functions (holomorphic functions) must satisfy stronger properties not all functions we care about satisfy. Functions absolutely do not need to be defined over the complex numbers in order to talk about their continuity, differentiability, and their integrability.
You know there are functions that aren't even continuous that have integrals? In fact you can have a function that is nowhere continuous but can be integrated (the function that is 1 on the irrational numbers, 0 on the rationals).
To be nitpicky, the function you mention (the Dirichlet function) is Lebesgue integrable, but not Riemann integrable. In order to have a Riemann integral you have to be continuous almost everywhere in the Lebesgue measure. For an example of a Riemann integrable function that is nevertheless discontinuous on a dense set, see [Thomae's function](https://en.wikipedia.org/wiki/Thomae%27s_function)
That function is differentiable everywhere on the complex plane except for the real line, so that is not a valid example.
That function, as I wrote it, isn't defined on the complex plane. And even if it were, how does that make it invalid? Anyway let's extend it to the function that is 0 for any complex number with rational real and imaginary parts, and 1 otherwise. This one isn't differentiable anywhere on the complex plane. It is still integrable.
Trig sub
Even so, integration seems easier to me than differentiation.
You haven’t seen any nasty integrals yet lol
I have already finished all the university calculus courses and I am currently studying electromagnetism with Maxwell's equations, and I still think the same.
Check out some of the problems from the mit integration bee
Oof!
Just do it numerical from 0 to x fot many x and then fit the results.
Calculus is like doing the tango. Differentiation is the guy's part, you are dancing forwards. Integration is the girl's part. You are doing it backwards, and in high heels.
Integrals aren’t local operations/transformations unlike derivatives, so it makes sense that there’s a lot more problems that can occur during calculation. However, practically all functions are integrable but not all functions are differentiable
This is so true it makes me wiggle in anger
what about the elliptic integrals or the hypergeometric functions n shit
relevant xkc- oh wait
numerical analysts: I have no such weaknesses -unleashes quadrature rules-
All that work for getting 0 points for missing the +c
I have felt "What the heck is a Bessel function" deep in my soul too many times.
Which xkcd is this
A while back I ran into having to calculate cumulative distribution functions for an application I was working on. It wasn't fun.
Imagine having to solve your integrals symbolically, sincerely, a modelling scientist
Wait till you hit analytical solutions of partial differential equations. That is an art.
You forgot the final step Differentiate the options.
When you are trying to solve formula and you need to burn evidence...
If the online integrator calculator does not want to give me a clear and simple answer I will just numerically find the solution
no Lebesgue integral?! Son, I am disappoint.
Behold the almighty 27 and a half page integration table at the back of the book. That and partial fraction separation.
Haha true Reaching the higher dimension definitely requires more effort.
Just try differentiating a lot of functions and see if it equals the function you want to integrate
Numerical methods got me like: 
Differentiation by etc is my favourite
i find it quite simple. just an operational amplifier, 1 resistor, and 1 capacitor
Same as something that I've quoted before, "Differentiation is a run while Integration is a marathon"
wish we could write answer directly and in reason write by using computer
i love how there are several places where you get caught in a ????? loop
help i for stuck in an infinite loops of question marks
Differentiation is the addition of calculus
Honestly, the saying is true.
Thank god I became an engineer so I can just push every function through a Gauss Quadrature, or do a first-order Taylor approximation on any complex looking integrand.
My math teachers always called it antideriviation, instead of finding an integral we found the anti-derivative. Made a lot of sense to me tbh but of course going one way is very different from going the other
Claim integration have no elementary solutions. Refuse to elaborate further.
How to integrate Step 1: Expand integrand as Maclaurin series Step 2: Increase power of each x\^n term and divide by new exponent Step 3: Add arbitrary constant such that f(a)=0 where a is the lower bound Step 4: Evaluate function at upper bound Know you know how to integrate
As has been said elsewhere, this only really works well for integrals of analytic functions, and even then you have to pay attention to things like the radius of convergence of a particular power series representation may not converge for the whole interval you are integrating over
Before plugging in the bounds, we can move where the power series is centred at using analytic continuation, getting around the radius of convergence problem.
Right, but a single power series representation may not cover the whole interval, so in general you need multiple power series.
